René Thomas

Regulatory networks seen as asynchronous automata: A logical description

The aim of this paper is to provide a compact answer to the questions: why treat complex biological systems in logical terms? how can one do it conveniently? Our initial description (Thomas, R. J. theor. Biol. 1973, 42, 563) is what we now call the “naive” logical description. After recalling the essential elements of this asynchronous description, the present paper introduces - the use of logical variables with more than two values - the notion of logical parameters - the logical identification of all steady states of the differential description - the notion of “characteristic” state of feedback loops - a compact matricial presentation This is an essentially methodological paper. More extended developments including concrete biological examples will be found elsewhere ( Thomas & DAri, 1990 ).

POSITIVE LOOPS AND DIFFERENTIATION

This paper focuses on the relation between the presence of positive feedback loops and the occurrence of multiple states of gene expression. After a short recall on single feedback loops and their properties, we discuss more extensively the properties of positive loops. This discussion includes a theorem (demonstrated elsewhere) which states that the presence of positive loop(s) is a necessary condition for multistationarity. We also discuss some general principles for pattern formation, in terms of involvement of different types of positive feedback loops. Finally, we briefly mention recent experimental results involving positive loops in crucial differentiative processes.

Passive replication of heteroimmune bacteriophage in a λ-driven replicon

Abstract Thermal induction of λ c 857 -434 hy and λ c 857 -80 hy tandem dilysogens results in replication not only of the λ, which is induced, but also of the other phage. Nevertheless, the latter remains under immunity control, since: (a) a superinfecting phage of that immunity fails to replicate; (b) the genes under direct immunity control are not expressed. Replication of the non-induced phage takes place only if it is adjacent to the λ, and only if the λ itself is able to replicate: however, this replication does not require that the replicative enzymes are interchangeable between λ and the other phage. Clearly, one deals here with passive replication of a phage, as a part of a λ-driven replicon. Function Int frequently excises the whole tandem of prophages, presumably as a double length circle. Replication from this structure, followed by the action of function Ter, accounts for our results as well as for those of Whitfleld & Appleyard (1958). The situation described here is in many respects similar to that analyzed by Gottesman & Yarmolinsky (1968). In their case, however, passive replication takes place in situ because int − mutations in the prophages prevent early excision, and particles of only one immunity are found in the yield.

THE RÖSSLER EQUATIONS REVISITED IN TERMS OF FEEDBACK CIRCUITS

We would like to show that complex dynamics can be deciphered and at least partly understood in terms of its underlying logical structure, more specifically in terms of the feedback circuits built in the differential equations. This approach has permitted to build a number of new three- and four-variable systems displaying chaotic dynamics. Starting from the well-known Rossler equations for deterministic chaos, one asks how systems with the same types of steady states can be synthesized ab initio from appropriate feedback circuits (= appropriate logical structure). It is found that, granted an appropriate logical structure, the existence of a domain of chaotic dynamics is remarkably robust towards changes in the nature of the nonlinearity used, and towards those sign changes which respect the nature (positive vs negative) of the feedback circuits. Using logical arguments, it was also easy to find related systems with a single steady state. A variety of 3- and 4-d systems based on other combinations of feedback circuits and generating chaotic dynamics are described. The aim of this work is to contribute to a better understanding of the respective roles of feedback circuits and nonlinearity — both essential — in so-called non trivial behavior, including deterministic chaos. Special emphasis is put on the interest of using the Jacobian matrix and its by-products (characteristic equation, eigenvalues, …) not only close to steady states (where linear stability analysis can be performed) but also elsewhere in phase space, where precious indications about the global behavior can be collected.